We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measure-preserving map is established by formulating the problem in the context of optimal transport theory. We discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem, exploiting gradient information from the forward model when possible. The resulting algorithm overcomes many of the computational bottlenecks associated with Markov chain Monte Carlo. Advantages of a map-based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves. The optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood. We demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension, involving the inference of parameters appearing in ordinary and partial differential equations.
Abstract- I. INTRODUCTIONThe integration of RF, analog and digital circuitry on a single integrated-circuit substrate, or system-on-chip (SoC), is associated with many challenges. For instance, substrate noise may lead to altered circuit performance. In addition, conductor skin and proximity effects may impact current return paths in a network of closely-spaced conductors, as well as quality factor in RF inductors. Moreover, the occurrence of radiated electromagnetic interferences (EMI) may transmit disturbances across different SoC systems. In order to avoid all the above problems, electromagnetic simulation tools are needed that can accurately and efficiently extract wide-band full-wave conductor impedances in the presence of semi-conducting substrates.In the last decade, electromagnetic simulators based on integral formulations have been the subject of extensive research due to the development of fast algorithms i.e. O (N log N) In this paper we present a 3D full-wave simulator based on the mixed potential integral equation formulation (MPIE) [7], [8], and accelerated using p-FFT. The solver will be released on public domain starting from December 2006 [9].
We present the fundamentals of a measure transport approach to sampling. The idea is to construct a deterministic coupling-i.e., a transport map-between a complex "target" probability measure of interest and a simpler reference measure. Given a transport map, one can generate arbitrarily many independent and unweighted samples from the target simply by pushing forward reference samples through the map. If the map is endowed with a triangular structure, one can also easily generate samples from conditionals of the target measure. We consider two different and complementary scenarios: first, when only evaluations of the unnormalized target density are available, and second, when the target distribution is known only through a finite collection of samples. We show that in both settings the desired transports can be characterized as the solutions of variational problems. We then address practical issues associated with the optimization-based construction of transports: choosing finite-dimensional parameterizations of the map, enforcing monotonicity, quantifying the error of approximate transports, and refining approximate transports by enriching the corresponding approximation spaces. Approximate transports can also be used to "Gaussianize" complex distributions and thus precondition conventional asymptotically exact sampling schemes. We place the measure transport approach in broader context, describing connections with other optimization-based samplers, with inference and density estimation schemes using optimal transport, and with alternative transformation-based approaches to simulation. We also sketch current work aimed at the construction of transport maps in high dimensions, exploiting essential features of the target distribution (e.g., conditional independence, low-rank structure). The approaches and algorithms presented here have direct applications to Bayesian computation and to broader problems of stochastic simulation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.